President Obama, reacting to the brutal execution of an American journalist by Islamic State terrorists, said Wednesday that the militant organization has "no place in the 21st century," as U.S. intelligence analysts confirmed the authenticity of a video showing the journalist's beheading.

The president, in brief remarks from Martha's Vineyard, said the video of James Foley's death "shocks the conscience," while vowing to continue the fight against the organization which has taken root across swaths of Iraq and Syria. He called for a "common effort" across the Middle East to "extract this cancer so it does not spread."

"The United States of America will continue to do what we must do to protect our people," Obama said. "We will be vigilant and we will be relentless."

Military analysts and U.S. lawmakers also say the execution should be a wake-up call to the need to dismantle the militant organization.

The remarks come after the White House National Security Council released a brief statement saying the intelligence community reviewed the video and has "reached the judgment that this video is authentic." Obama said he spoke with Foley's parents earlier Wednesday.

The beheading marks the first time the Islamic State -- also known as ISIS, or ISIL -- has killed an American citizen since the Syrian conflict broke out in March 2011, upping the stakes in an increasingly chaotic and multilayered war.

“America got a glimpse of exactly who they are,” House Intelligence Committee Chairman Mike Rogers, R-Mich., told Fox News on Wednesday. “This is a group you need to deal with.”

Sen. Marco Rubio, R-Fla., called the "brutal execution" of a hostage "the clearest indication to date that ISIL has declared war on the United States."

The militants in the video railed against U.S. airstrikes in Iraq and threatened to kill a second American journalist, depending on Obama’s “next decision” – an apparent reference to U.S. military policy in the region.

The video puts pressure on the White House to weigh the risks of ramping up airstrikes in Iraq targeting the Islamic State, and the possibility that militants could kill more Americans in response.

But in an indication that the U.S. response is not changing, a U.S. official said American fighter jets and drones conducted nearly a dozen strikes in Iraq since Tuesday. The official told the Associated Press the airstrikes were in the area of the Mosul Dam and were aimed at helping Iraqi and Kurdish forces create a buffer zone at the key facility.

Rogers and others said the video only underscores the need to develop a clear strategy for defeating the group. Rogers said this week’s success, for instance, in helping Iraqi and Kurdish forces retake the Mosul Dam from militants must be part of a “bigger strategy.”

“That [video] ought to get us off our backsides and get to work on dismantling this organization. It’s dangerous,” he told Fox News.

“We have to be taking severe military action,” Lt. Col. Anthony Shaffer, a former military intelligence officer now with the Center for Advanced Defense Studies, told Fox News, saying “boots on the ground” are not necessarily needed to do that.

“We do have the ability to start walking back ISIS, we can't wait,” he said, echoing lawmakers’ concerns that the group will try to attack New York City, Washington and other American targets if they can.

Retired Gen. Michael Hayden, former director of the CIA and National Security Agency, raised the same issue.

“What we've seen with this video is the character of the Islamic State. Do you want that to continue to exist and thrive and grow and be able to threaten other Americans?” he said on Fox News.

Foley's family also confirmed his death in a statement posted on a Facebook page that was created to rally support for his release, saying they "have never been prouder of him."

"He gave his life trying to expose the world to the suffering of the Syrian people," said the statement, which was attributed to Foley's mother, Diane Foley. She implored the militants to spare the lives of other hostages. "Like Jim, they are innocents. They have no control over American government policy in Iraq, Syria or anywhere in the world."

Foley, 40, from Rochester, N.H., disappeared in northern Syria in November 2012. He had previously reported for several outlets, including Stars and Stripes, but was freelancing for Agence France-Presse and the Boston-based media company GlobalPost at the time. The car he was riding in was stopped by four militants in a contested battle zone that both Sunni rebel fighters and government forces were trying to control. He had not been heard from since.

The group is the heir apparent of the militancy known as Al Qaeda in Iraq, which beheaded many of its victims, including American businessman Nicholas Berg in 2004.

The video released on websites Tuesday appears to show the increasing sophistication of the Islamic State group's media unit and begins with scenes of Obama explaining his decision to order airstrikes.

It then cuts to a balding man in an orange jumpsuit kneeling in the desert, next to a black-clad militant with a knife to his throat. Foley's name appears in both English and Arabic graphics on screen. After the captive speaks, the masked man is shown apparently beginning to cut at his neck; the video fades to black before the beheading is completed. The next shot appears to show the captive lying dead on the ground, his head on his body. The video appears to have been shot in an arid area; there is no vegetation to be seen and the horizon is in the distance where the sand meets the gray-blue sky.

At the end of the video, a militant speaking with a British accent shows a second man, who was identified as another American journalist, Steven Sotloff, and warns that he could be the next captive killed. Sotloff was kidnapped near the Syrian-Turkish border in August 2013; he had freelanced for Time, the National Interest and MediaLine.

New Hampshire lawmakers condemned the killing. Sen. Jeanne Shaheen, D-N.H., called it a "cowardly act of terrorism" that "underscores the threat that ISIL poses to the freedoms we hold dear."

In a statement Tuesday evening, Rep. Adam Schiff, D-Calif., who sits on the House Intelligence Committee, also called the act "appalling."

“Seldom is the descriptor 'evil' applied with perfect accuracy as it is with this monstrous group that glories in death," he said. "They know no human decency -- murdering journalists, beheading religious minorities refusing to convert, victimizing women and children, and starving entire communities.”